A theory for the relationship between the fundamental single-walled carbon nanotube structure parameters (diameter and chirality) and the optical transitions of absorbance and emission (photoluminescence) of light energy is known from the work of Weismann and coworkers (see e.g. Bachilo et al. (2002) Science 298:2361-2366).
SWCNTs are formed by a vectorial wrapping of a sheet of graphite crystal into a cylinder. The parameters defining the direction of the wrapping vector comprise what is known as the zigzag axis and the armchair axis, where the potential ‘chirality’ angle of the ‘wrapping vector’ is constrained between 0 degrees and 30 degrees between the two axes. The diameter of the tube is defined by the length of the wrapping vector and the chirality angle. The graphite sheet is mapped with two integer parameters, n and m, which plot the coordinates from n,m having the values (0,0) (the coordinate at the origin of the wrapping vector) to n,m having the values (x,y), where x and y are the coordinates marking the terminus of the wrapping vector. As alluded to above, the angle defined from (0,0) to (x,y) is defined as the chirality angle. Hence, a carbon nanotube can be equivalently described by its diameter (dt) or by its chirality angle and terminal (n,m)=(x,y) values.
Bachilo et al., above, disclose a mathematical relationship between the size and chirality of a given nanotube species and the energy values for both the absorption and emission optical transitions. Bachilo et al. define equation systems used to describe the observed relationships between the SWCNT diameter parameter (dt) and two key optical transition parameters, namely 1) the semiconductor absorption or upward transition of electrons from the valence band 2 to conductance band 2, i.e. the energy of a photon which drives an electron across the bandgap when it is excited, and 2) the subsequent emission transition of electrons from the conductance band 1 to the valence band 1, i.e. the energy of a photon emitted by the carbon atom when an electron in the conductance band 1 returns across the bandgap to the valance band 1.
Essentially, the simplest, theoretical relationships can be viewed as quasi-linear according to the following equations:dt=λ224accγ0/hc=λ112accγ0/hc Where, λ22 is the photon energy needed to drive the electron in the transition from v2 to c2 and λ11 is the energy of the photon emitted during the transition from c1 to v1, acc is the C—C bond distance and γo is the interaction energy between neighboring carbons, h is Planck's constant and c is the speed of light.
Bachilo et al (2002) used a HORIBA Jobin Yvon Fluorolog spectrophotometer equipped with a near infrared detector to generate emission intensity measurements and illustrated them as a topography of intensities for a matrix of excitation wavelengths and emission wavelengths. More particularly, Bachilo et al. described a technique in which photon energies λ22 and λ11 are measured over a range of excitation and emission wavelengths, and peaks in the emissions are identified for use in the above equations to solve for diameter and chirality. These peaks may be visualized as three-dimensional surface with peaks defining emission peaks, and valleys between and on the sides of the peaks.
As shown by Bachilo et al, however, the observed correlations do not fit the predictions of the above equations robustly. A more thorough model parameterization was invoked to take into consideration the chirality and other structural features of the SWCNTs species and families.
The general conclusion accepted by most authorities today is that the best fitting relationships between λ11 and λ22 are described using the following model equation system where the frequencies of the optical transitions in reciprocal centimeters are:
                              v          11                =                                                            1                ⨯                                  10                  7                                            ⁢                              cm                                  -                  1                                                                    157              +                              1066.9                ⁢                                                                  ⁢                                  d                  t                                                              +                                                    A                1                            ⁢                              cos                ⁡                                  (                                      3                    ⁢                    α                                    )                                                                    d              t              2                                                          (        1        )                                          v          22                =                                                            1                ⨯                                  10                  7                                            ⁢                              cm                                  -                  1                                                                    145.6              +                              575.7                ⁢                                  d                  t                                                              +                                                    A                2                            ⁢                              cos                ⁡                                  (                                      3                    ⁢                                                                                  ⁢                    α                                    )                                                                    d              t              2                                                          (        2        )            where α is the chirality angle and A1 and A2 are specific constants (710 cm−1 and 369 cm−1, respectively) referring to families of SWCNTs related by parallels in their n and m coordinates, and where:ν11 cm−1=(1/λ11 nm*107 nm/cm)  (3)andν22 cm−1=(1/λ22 nm*107 nm/cm)  (4)
In theory, one could measure the absorption and emission matrices of SWCNT mixtures and use the above equations to determine the diameter and chirality of SWCNTs in a given mixture after the peak excitation-emission coordinates have been observed. This is done by simultaneously solving equations (1) and (2) for d1 and α and substituting values for ν11 and ν22 calculated using equations (3) and (4).